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Multivariate Statistics

ESM 206, 5/17/05

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WHAT IS MULTIVARIATE STATISTICS?

• A collection of techniques to help us understand patterns in and make predictions with large datasets with many variables

• Ordination: find a (hopefully small) number of composite variables that capture most of the variability among data points

• Cluster Analysis: discover natural groupings of similar data points

• Discriminant Analysis: find a (hopefully small) number of composite variables that can be used to predict the levels of a categorical dependent variable

• Canonical Correlation Analysis: find relationships between two groups of variables– “Dependent variable” is multivariate

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WHAT CAN MULTIVARIATE STATISTICS DO?

• Reflect more accurately the true multidimensional nature of environmental systems

• Provide a way to handle large datasets with large numbers of variables by summarizing the redundancy

• Provide rules for combining variables in an “optimal” way

• Provide a means of detecting and quantifying truly multivariate patterns that arise out of correlational structure of the variable set

• Provide a means of exploring complex data sets for patterns and relationships from which hypotheses can be generated and subsequently tested experimentally

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DISTINGUISHING ECOLOGICAL NICHES OF 3 SPECIES

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ORDINATION

• Simplify the interpretation of complex data by organizing sampling entities along independent gradients or factors defined by combinations of interrelated variables

• Uncover a more fundamental set of factors that account for the major patterns across all of the original variables

• If a few major gradients explain much of the variability in data, then data can be interpreted with respect to these gradients without loss of information

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PRINCIPAL COMPONENTS ANALYSIS (PCA)

• Most commonly used ordination technique

• Given P correlated variables, extract P principal components– Linear combinations of the variables

– Uncorrelated with one another

– First PC is direction through data cloud that captures the most variance in data

– Second PC is direction perpendicular to first that captures the most remaining variance

– Etc.

• Assumptions of PCA:1. Data are multivariate normal

2. Data are independent

3. Observed variables depend linearly on underlying factors

• May need to transform data to satisfy these

• Unless variables are all measured on same scale, use correlations rather than covariances– Gives equal weight to variability in

all variables

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EXAMPLE: CHEMICAL SOLUBILITY

• 72 chemical compounds tested for solubility in each of 6 solvents– Solubility measure on log scale

• Strong (but not perfect) correlations among the 6 solvents

• Can we use fewer than 6 variables to characterize each chemical?

-0.5

0.5

1.5

2.5

-1.00.01.0

2.5

-1

1

3

-2

0

2

-3

-1

1

3

-4

-2

0

2

1-Octanol

-0.5 .5 1.5 2.5

Ether

-1.0.0 1.0 2.5

Chloroform

-1 0 1 2 3 4

Benzene

-2-1 0 1 2 3

CarbonTetrachloride

-3 -1 0 1 2 3

Hexane

-4 -2 0 1 2 3

Scatterplot Matrix

Multivariate

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Eigenvalue

Percent

Cum Percent

1-Octanol

Ether

Chloroform

Benzene

Carbon Tetrachloride

Hexane

Eigenvectors

4.7850

79.7502

79.7502

0.37441

0.34834

0.41940

0.44561

0.43102

0.42217

0.9452

15.7538

95.5040

0.55987

0.64314

-0.29864

-0.14756

-0.29736

-0.27117

0.1399

2.3309

97.8348

-0.11070

0.11973

-0.64850

-0.21904

0.18487

0.68608

0.0611

1.0182

98.8530

-0.65842

0.62764

0.30599

-0.09455

-0.24135

0.10831

0.0471

0.7850

99.6380

0.31660

-0.20890

0.43061

-0.49849

-0.45965

0.45926

0.0217

0.3620

100.0000

0.01874

0.11456

0.18793

-0.68865

0.64968

-0.23426

Principal Components: on Correlations

Principal Components / Factor Analysis

Multivariate

SOLUBILITY PCA• Eigenvalue indicates how much of

the variability in data is explained by the PC – Magnitude depends on number of

variables (and variances if done with covariance matrix)

– Instead look at percents

• Eigenvector gives coefficients of linear relationship of PC to each variable– NOTE: some software scales the

eignvectors differently

• Interpretation: PC1 is axis of overall increasing solubility

• PC2 is axis of differential solubility in 1-Ocatanol & Ether vs. other 4 solvents

PC1 0.37 0.35 0.42 0.45

0.43 0.42

O E C B

CT H

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-2

-1

0

1

2

PC

2 (p

refe

rent

ial s

olub

ility

by 1

-Oct

anol

& E

ther

)

-6 -5 -4 -3 -2 -1 0 1 2 3 4 5 6

PC1 (overal solubility)

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CHARACTRISTICS OF ORDINATION• Organizes sampling entities (e.g.,

species, sites, observations) along continuous environmental gradients

• Assesses relationships within single set of variables; doesn’t define relationship between a set of independent variables and one or more dependent variables– However, PC’s can be used as

independent variables in a regression

• Reduces dimensionality of multivariate data set by condensing large # of original variables into smaller set of new composite variables with minimal loss of information

• Summarizes data redundancy by placing similar entities in proximity in ordination space

• Defines new composite variables (e.g., principal components) as weighted linear combinations of the original variables

• Eliminates noise from a multivariate data set by recovering patterns in first few composite dimensions and deferring noise to subsequent axes

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OTHER ORDINATION TECHNIQUES

• Polar Ordination (PO)• Factor Analysis (FA)

– This is often used as a generic term meaning “ordination” in social sciences

• Nonmetric Multidimensional Scaling (NMMDS)– Relaxes normality and linearity

assumptions by using ranks

• Correspondence Analysis (CA)– Allows data (e.g., species

abundance) to take on peak values at intermediate levels of the gradient

– Also called Reciprocal Averaging

• Detrended Correspondence Analysis (DCA)– Deals particularly well with

nonlinear relationships

• Canonical Correspondence Analysis (CCA)– Like CA, but ordination of variables

of interest (e.g., species abundance) is constrained to depend linearly on other variables (e.g., environmental characteristics) measured at same sites

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FURTHER READING

• McGarigal, K., S. Cushman, and S. Stafford. 2000. Multivariate Statistics for Wildlife and Ecology Research (Springer-Verlag, New York).

• Gotelli, H.J., and A.M. Ellison. 2004. A Primer of Ecological Statistics (Sinauer, Sunderland, MA); Chapter 12.

• Spicer, J. 2005. Making Sense of Multivariate Data Analysis (Sage Press, Thousand Oaks).